For analog IC design engineers, using matched components in circuits is one of the keys to creating highly accurate circuits. I have worked on many types of power-management chips, and a small, yet very important, part of them would be the matched feedback resistors that contribute to output voltage accuracy.
To optimize the matching of resistors on an IC, you would typically use multiple segments of the exact same size resistors. An advantage to this approach is that process variations, contact resistances, and geometries will stay relatively consistent in every resistor. Therefore, the voltage divider always produces the same ratio.
However, sometimes the designer need only use a fraction of one of these unit resistors. As we all learned back in EE 101, if you need half of a unit resistor, it's easy enough to connect two unit resistors in parallel to produce half the resistance. Consequently, if you need one-third of a unit resistor, three in parallel will work, as the pattern continues.
But what if you need 3/5 or 4/11 of a unit resistor? Admittedly, the more obscure fractions aren't often required, but this technique will work for any particular fraction.
Let's take the example of 3/5 of a unit resistor. The brute-force method of producing 3/5 of a unit resistor is to have a set of five resistors in parallel and put three of those sets in series (Fig. 1). This does work, but it uses 15 unit resistors. To conserve space, you probably want to keep the number of resistors to a minimum, so there's an obvious drawback to this approach.
To simplify the process, divide the resistors into square sets. Starting from the left, circle a set of 3-by-3 resistors and replace it with an equivalent one-unit resistor (Fig. 2a). Repeat the process with the N-by-N resistors that remain until you can do no more (Fig. 2b). The result: What started out as 15 resistors quickly boils down to four.
Figure 3 shows the same method for 4/11 of a unit resistor. Try it with other fractions. It may take a little practice to find the optimum N-by-N resistors to simplify, but it's fairly easy to get the hang of this technique—one that may help save a little time on your next chip.
